(Microsoft PowerPoint - La complessit\340 della realt\340 biologica

La complessità della realtà
biologica interpretata tramite
la geometria frattale
Gabriele A. Losa Prof. PhD
European Academy of Sciences
Chairman Fractals in Biology and Medicine
Institute of Scientific Interdisciplinary Studies [ISSI]
Locarno
Switzerland
e-mail: [email protected]
POLITECNICO DI MILANO
Anno accademico 2013/2014
6° ciclo di seminari FDS
Aula Consiglio del Dipartimento di Matematica
4 dicembre 2013 ore 15.00
Le forme biologiche
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Stato attuale: il problema della forma costituisce il nocciolo centrale nella
comprensione ed elucidazione dei processi biologici e genetici .
La biologia risulta dall’interazione (interplay) fra:
caso (hasard, randomness): ruolo nell’evoluzione
non linearità (nonlinearity): creazione delle forme
complessità (complexité, complexity): generazione, modellatura (adattamento)
(modelage, adaptation) delle forme .
Studio delle forme biologiche:
Leonardo da Vinci : « Nessuna humana investigatione si può dimandare vera
scienza s’essa non passa per le matematiche dimostrazioni >. (MSA 100): …. la
natura,…. la quale con filosofia e sottile speculazione considera tutte le qualità delle
forme, …e siti, piante, animali, erbe e fiori, le quali sono cinte d’ombra e di luce. E
veramente questa è scienza ….
Galileo Galilei : « l’Universo è scritto in lingua matematica, e i caratteri son
triangoli, cerchi ed altre figure geometriche.. «
D’Arcy Thompson: « En général les formes n’existent pas, sauf celles qui sont
conformes aux lois de la physique et de la mathématique.
E. Kant : E. Kant : The organization of forms is an abyss impenetrable to the mind.
Ma quale linguaggio ?
Convenzionale, matematico analitico, geometria frattale, altro ?
Mandelbrot : ‘‘. . clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in
a stright line “.
The Irruption of Fractal Geometry in Biology and Medicine
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Although there is no precise date, it is generally agreed that the introduction of Fractal Geometry into the
life sciences, such as biology and medicine, occurred within the ‘‘golden age’’ of cell biology – that is,
between the 1960s and 1990s.
There was a striking debate within the biologist community, about the morphofunctional complexity of
cells and tissues and how to measure it.
The discovery that cellular membrane systems have fractal properties arose from the uncertainty of
observations regarding the extent of such membranes. When the results of the first studies on the
morphometry of liver cell membranes were reported, the values obtained were much higher than had been
reported by others; in fact liver cells were found to contain 6 or 11m2 of membranes per cm3 . Which one
of these estimates was correct ?
The systematic measurement of irregular liver cell membranes revealed that the estimates of surface density
increased with increased resolution [Paumgartner et al.]. Shortly after the conclusion of the experimental
study, Mandelbrot suggested that the results should be interpreted with the likely effect of the ‘‘resolution
scale,’’ in analogy with the ‘‘Coast of Britain effect’’ . (Not Euclidean linear bodies).
Paumgartner, D., Losa, G.A., Weibel, E.R. (1981). Resolution effect on the stereological estimation of surface and volume
and its interpretation in terms of fractal dimensions. J. Microsc. 121, 51–63.
Mandelbrot, B. (1967) How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science 155,636–
640.
Scale length effect
Main Properties of Fractal Elements
• Self-similarity [geometrical or statistical]
• Form invariance by scaling
• Shape irregularity (rugosity)
• Iteration of the generator
• Fractal dimension FD (fractional/non-integer)
• Mandelbrot-Richardson equation: scaling power law
log L(ε) = (1-D) log (ε)
lo
lo
Remark: The complexity and the self-similarity of biological shapes are
constructed from inherent simplicity (uncomplicated rules), i.e. by the
iteration of an invariant simple pattern that spans several length scales.
Fractal Geometry
Unifying description of not uniform natural phenomena and structures
Mathematical fractals
exact self-similar, invariant over an unlimited range of scales (ε)
Biological (natural) fractals
• statistically self-similar
• invariant within a finite range of ε values - scaling window - defined by
upper and lower limits (ε1: ε2) on a log-log plot
spreading at least two orders of magnitude.
• measured properties (functional/ morphological/ temporal) depend on scaling
(i.e. on magnification / scale of observation)
• high level of organization (complexity)
Mathematical basis
Mandelbrot-Richardson equation
and
L(ε) = N(ε). (ε )
N(ε) = l0D ε-D
L(ε) = [ ε ] 1-D
lo
[lo]
The logarithmic transformation yields a straight line equation
log L(ε) = (1-D) log (ε)
lo
lo
Dimensionless Scaling Power Law
Fractal dimension [D]: estimated from the slope (1-D) of the straight line
drawn on a log-log plot
The fractality of natural objects
The Scaling Window
A typical N versus ɛ plot (log-log scale) based on the power law N(ɛ) = ɛ (1-D)
The scaling window is clearly indicated: Ɛ1(0.025 mm) – Ɛ2 (25 mm).
The slope of the straight line which could be plotted within this window is used to calculate
the fractal dimension [D]. Box-counting method: the ordinate N (number of boxes)
increases as the pixel length (Ɛ) decreases, showing that the outline length of the object
examined also increases within the scaling interval Ɛ1-Ɛ2.
Ex.: NMBHC from MCF-7 breast cancer cells treated with 10-9 M estradiol.
Mitochondria
Mitochondria 130
Surface density estimates for outer and inner mitochondrial membranes
Paumgartner et al. J. Microscopy 1981;121:51
Resolution scale
Normal Breast Tissue / Invasive Carcinoma
CD-8 T-cell
T-ALL
Fractal dimension of lymphocytes and leukemic cells
Steroid hormones on nuclear heterochromatin (MCF-7 cells)
FD of nuclear membranes
Apoptosis / Programmed Cell Death
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Physiological form of Cell Death (opposite: necrosis)
Control of adult tissues and regulation of embryonal
development
Balance between cell proliferation and cell number
Defense mechanism against damaged, dangerous,
and infected cells
Elimination of cells with altered or endomaged DNA, with
mutations or initiatiors of cancers
Elimination of exhausted cells during tissue and organ
developpment ( nervous system, immune system, ..)
Regulation of cell associations within tissues
Apoptosis of SK-BR-3 cells
Apoptotic markers
FD of interchromatin space in early apoptotic SK-BR-3 nuclei
Fractal tree model
Oocyte with and without Cumulus Oophorus [COC]
Oocyte grey-dark cytoplasm texture
with COC
without COC
Fractal dimension: methodological comparison
Oocytes with intact COC and successively denuded of COC using finely-drawn glass capillary pipette. A binary image was obtained by
grey tresholding the area occupied by the grey-dark of the oocyte cytoplasm (B-B1); two different outlines were extracted from this
binary image by applying a Roberts filter: one pertained to the internal texture of the grey-dark cytoplasm area as well as to the
scattered grey-dark particles within the cytoplasm (C-C1 ), while the other referred to the external profile only of the grey-dark
cytoplasm (D-D1). Original images of each oocyte were acquired using a 40x objective lens.
Conventional morphometry : tissue percentage distribution
Fractal geometry: irregularity and true surface/ interface
Vv= 100 x Pe/Pt ; Vv = 100 x Pm/ Pt.
A
B
C
A) Canine Trichoblastoma: the epithelial component is positive for Cytokeratin, 4X.; B) Binary image
obtained by grey thresholding the area occupied by the epithelial component and C) outline of the area
occupied by the neoplastic epithelium.
Fractal dimension derived from masks
and outlines of canine trichoblastoma
RT
TT/GC
ST
Mask
Mean
Sd
1.75**
0.02
1.85**
0.02
1.78**
0.03
Outlines
Mean
Sd
1.70*
0.02
1.77 n.s.
0.02
1.76n.s.
0.02
RT, trichoblastoma of the ribbon type
TT/GCtrichoblastoma of the trabecular and Granular
Cellular
ST, trichoblastoma of the spindle
* * significant different at p< 0.001 from each other.
* RT significant different at p< 0.001 from TT/CG and ST .
n.s.: TT/CG ; ST, not significant from each other.
M outh- E C T interface
Irregularity (by FD) of Epithelial connective tissue interface [ECTI]
Ageing of the oral mucosa does not seem to affect significantly the irregularity of the epithelial connective tissue interface [ECTI] as documented by
close FD values in the table below. A: Image of the epithelium. B: Haematoxylin component of A. C: segmented epithelial compartment. D:segmented
epithelial compartment after logical operation between A and C. Magnification 20X.
Mean Box fractal dimension and the number of cases for different age groups.
Age Range
Mean Box Fractal Dimension
Number of cases
0-10
1.1069 ± 0.03267
2
11-20
1.1117 ± 0.08987
2
21-30
1.1414 ± 0.07095
4
31-40
1.1290 ± 0.03822
4
41-50
1.0933 ± 0.05661
5
51-60
1.1373 ± 0.05657
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61-70
1.0992 ± 0.04456
9
71-80
1.1260 ± 0.04751
5
81-90
1.0903 ± 0.01675
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Fractal analysis of contours of breast masses in mammograms
37 benign breast masses and 20 malignant tumors were ranked by their fractal
dimension FD estimated by the 1D ruler method.
Rangayyan et al. J Digit. Imag. 2007: 220: 223
Brain and nervous system: a brief chronicle
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Galeno (Pergamo) 129-216 d.C.: Conception : superior functions
within three cerebral cells (spheres)
Leonardo 1452-1519: similar conception as Galeno
Andreas Vesalius 1543: De humani corporis fabrica . Cerebral
convolutions without identification of morphological pattern
Marcello Malpighi 1628: (crebral glands; nervous ( nerveo) fluid.
nerveo) .
Thomas Willis 1621:(arterial circuit by anastomosy of internal
carotids and vertebral artery.
Vicq d'Azyr:1748 :external surface of brain, convolutions ,
unidentified areas .
Albrecht von Haller 1708: secretive function of brain, nervous
fluid
Franz Joseph Gall / Johann Spurzheim-1810 : brain shape ---->
phrenologic maps with specific function.
Paul Broca 1861:localization of cerebral functions (langage :
"Nous parlons avec l‘ hémisphère gauche")
Carl Wernicke 1874: area of temporal lobe : its damage provokes
the sective loss of the capacity of listening words.
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Camillo Golgi (1843-1926) ramified nerve fibers could support the
'reticular theory', which postulated that the nervous system was a
syncytial system, consisting of nervous fibers forming an intricate
network, and that the nervous impulse propagated along such diffuse
network.
Human cerebellar cortex as drawn by Golgi (from the Opera Omnia).
Nervous staining technique: Black reaction: silver nitrate with
potassium dichromate- black deposit on the soma , axon and
dendrites .
Golgi apparatus: Golgi cells of Cerebellum: Golgi 1 ( nerve cells
with long axons); Golgi II ( nerve cells with short or no axons):
(GABA: γ-Aminobutyric acid, as trasmitter): inhibitor of CNS,
regulator of neuronal excitability and muscle tone
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Santiago Ramón y Cajal: (1852-1934) "neuron theory": the
relationship between nerve cells was not one of continuity, but
rather of contiguity. Dendritic spine: a small membranous
protrusion from a neuron's dendrite that typically receives input
from a single synapse of an axon (output).
Cerebellum
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Techniche di imaging ( tomografia ad emissione di positroni; NMR;
TAC;
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Analytical description and representation of biological elements
Mathematics/ statistics / Euclidean geometry/ conventional
morphometry
Fractal Geometry: morphological/ structural complexity:
power law scaling , self-similarity
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References:
M. Piccolino: Breve storia delle Neuroscienze
Losa GA, T. Nonnenmacher. Self-similarity and fractal
irregularity in pathologic tissues. Mod. Pathology 1996:9,174 .
Losa GA. Fractals in Biology and Medicine. Encyclopedia of
Molecular Cell Biology and Molecular Medicine: Systems
Biology. 2011, Wiley Press.
Andreas Vesalius (André Vésale)
The major portions of the Brain : Cerebrum, Cerebellum, Brain Stem
Brain complexity
1-month and 6-year-old-child
Increase in brain size and the maturation of cortical
circuits
Golgi stained cortical neurons of orbital gyrus
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J. De Felipe: The evolutionary concourse of two major events, “the tremendous expansion and the differentiation of the
neocortex,”, has contributed to the development of the human brain. J. Front. Neuroanatomy 2011, 5, 1-16
1982. In his masterpiece - The Fractal Geometry of Nature Mandelbrot, facing with the intricacy of mammalian brain folds, argued:
< A quantitative study of such folding is beyond standard geometry
but fits beautifully in fractal geometry >
At that time however, there was no certainty neither about the brain's geometry nor the neuron branching. The
anatomy-histological evidence that the complexity of plane-filling maze formed from dendrites of neural Purkinje
cells of cerebellum was more reduced in non mammalian species than in mammals led up Mandelbrot to comment:
< It would be very nice if this corresponded to a decrease in D (fractal dimension),
but the notion that neurons are fractals remains conjectural >
Since then, a wealth of investigations have documented the fractal organization of brain and nervous tissue system
while the implication of fractals for neurosciences was unambiguously assessed.
Pyramidal neuron
Fractal Self-Organization:Purkinje Cells in Cerebellum
Nervous fibers of white matter rebuild by IRMf (functional Resonance Magnetic Imaging)
The Geometric Structure of the Brain Fiber Pathways
The organizing principles of cerebral connectivity remain unclear and difficult to correlate with the structure of
brain. In the brainstem and spinal cord, fiber pathways are organized as parallel families derived from the three
principal axes of embryonic development: the rostro-caudal, the medio-lateral (or proximo-distal), and the
dorso-ventral. Several leading theories of cerebral function propose geometric organization at multiple scales.
However, high-resolution studies of cerebral connectivity with tract tracers have given only limited evidence of
geometric organization.(Van Weeden et al.2012.
NMR
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The neocortical column microcircuit
Markram H. The Blue Brain Project. Perspectives. Nature reviews 7, 2006
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b
c
d
Morphological development of thick-tufted layer V pyramidal
cells in the rat somatosensory cortex (conventional morphometry)
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Two major classes of layer V pyramidal neurons have been described . The first class
is termed the “thicktufted” (or “tall-tufted”) type (TTL5) ,characterized by a
thicktufted apical dendrite. Several oblique dendrites emerge from the main apical
dendrite (i.e., “apical trunk”) before bifurcating at a variable distance from the soma
to give rise to tuft dendrites, which usually branch extensively in layer I .
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Romand et al. Frontiers in Neuroanatomy 2011 ; 5 : 1-26
Neuronal images FD
FD ( box-counting method) of Dendritic Branching Pattern
Milosevic et al.. J.Theor. Biology 2009, 259, 142
Astrocytes 2
Astrocytes DS
Fractal analysis of astrocytes in stroke
and dementia
Figure 2. Typical illustration of 2D WM skeleton superimposed on an anatomical T1-weighted images.
Rajagopalan V, Liu Z, Allexandre D, Zhang L, et al. (2013) Brain White Matter Shape Changes in Amyotrophic Lateral Sclerosis
(ALS): A Fractal Dimension Study. PLoS ONE 8(9): e73614. doi:10.1371/journal.pone.0073614
http://www.plosone.org/article/info:doi/10.1371/journal.pone.0073614
Fractal dimension analysis of grey matter in multiple sclerosis (3D)
Grid classification for a 3D image composed of 156 2D slices of segmented grey matter with a threshold of 1 and for two
different box sizes: 2 (A) and 5 (B); due to the 3D representation, all boxes shown are classified as GREY. Boxes of sizes 2
(C) and 5 (D) for only one row of the total classification, where the box classification in BLACK (blue colour), WHITE (empty
colour) and GREY (green colour) is clearly shown. (E) Example of 3D fractal dimension computation through linear
regression over the box counting; the range selected for the linear regression is the set of green points (box sizes from 6 to
16);red points are the box counting of box sizes not selected for the final computation
Fractal dimension (FD) of the grey matter (GM) form controls and MS patients
Pellionisz, A. The Principle of Recursive Genome Function. The Cerebellum (Springer), 7(3) 348359,2008
Considerations and Hypothesis
Morphogenesis of biostructures follows fractal principles
Fractal geometry is a design principle for living organisms and may be envisaged for interpreting how
biological phenomena and shapes come about, while being well aware that the true reality may remain
undisclosed!
Do genes contain fractal algorithms?
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This appears much more interlocutory, because genes are DNA entities that codifyconstructive units or
templates, while fractal algorithms represent mechanisms (iteration, self-organization, environmental
constraints, etc.) which Nature may eventually adopt in order to assemble self-similar dynamic units into
final biological shapes.
Genomic/functional analysis, such as the RNA-interference (RNAi) technique adapted to plants and
mammalian cell cultures, has made it possible to screen systematically or genes controlling specific
cell-biological processes, including those required to influence cytoskeletal organization, to generate
distinct morphologies and, like Hox genes to act as architects which control the spatial structuring of the
organism through its entire development.
Fractal model of the Purkinje neuron (of the guinea pig), which links for the first time the fractality of the
genome with the fractality of neurons.The theoretical significance is that the fractality found in the DNA and
organisms for a long time “apparently unrelated” was put into a “cause and effect” relationship by the
Principle of Recursive Genome Function (Pellionisz 2008)